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Dissipative Partial Differential Equations and Dynamical Systems

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Dissipative Partial Differential Equations and Dynamical Systems Dissipative Partial Differential Equations and Dynamical Systems C.E. Wayne October 31, 2012 Abstract This article surveys some recent applications of ideas from dynamical sys- tems theory to understand the qualitative behavior of solutions of dissipative partial differential equations with a particular emphasis on the two-dimensional Navier-Stokes equations. 1 Introduction The focus of this article is the application of dynamical systems ideas to the study of dissipative partial differential equations. The notion of dissipativity arises in physics where it is generally thought of as a dissipation of some \energy" associated with the system and such systems are contrasted with energy conserving systems like Hamiltonian systems. In finite dimensional systems, the notion of dissipativity is relatively easy to quantify. If we have a system of ordinary differential equations (ODEs) defined on Rn, x_ j = fj(x) = fj(x1; : : : ; xn) ; j = 1; : : : n ; (1) then a common definition of dissipativity is that there be some bounded set, B, which is forward invariant under the flow defined by our differential equations and such that every solution of the system of ODEs eventually enters B,[Hal88]. Such a set is referred to as an absorbing set. If an absorbing set exists, and if φt is the flow defined by this dynamical system, then we can define an attractor for the system by t A = \t≥0φ (B) ; (2) which will be compact and invariant, and will have the property that any trajectory will approach this set as t ! 1. 1 Another property often associated with dissipativity is that the determinant of the Jacobian of the vectorfield in (1) is negative, i.e. @fj det f gj;k=1;:::;n < 0 : (3) @xk If this determinant is zero, then the system conserves phase space volumes, and this is one of the properties associated with Hamiltonian systems. Condition (3) means that the systems \dissipates" phase space volume, but note that it is not, in itself, sufficient to insure that we have a bounded attractor for the system, as the two dimensional example 1 x_ = x ; x_ = −x ; (4) 1 2 1 2 2 for which almost every solution tends to infinity shows. However, if the system satisfies (3), and is in addition dissipative in the sense described above, then we can immediately conclude that the attractor for the system has zero n-dimensional volume. Thus, the asymptotic behavior of the system is determined by what happens on a very \small" set. One important direction of research in the study of dissipative systems is to focus on the properties of the attractor. In special cases, the attractor may consist of a small number of simple orbits, like stationary solutions and their connecting orbits, or periodic orbits. In other, more complicated cases, the attractor may contain chaotic trajectories but it may itself live in a manifold of much lower dimension than the number of degrees of freedom of our original system. In such circumstances, the long-time behavior of arbitrary solutions of the original system can be determined from a study of the possibly much smaller system obtained by restricting the original ODEs to this manifold containing the attractor. This dimensional reduction has been a powerful tool in the study of dissipative systems. When one turns from ODEs to partial differential equations (PDEs), the discussion becomes more complicated due to the infinite dimensional nature of the problem. For one thing, the long-term behavior of the system may well depend on the norm we choose on our space of solutions. However, even after one has fixed the norm on the system the situation can be problematic due to the fact that closed and bounded sets need no longer be compact. Thus, even if we find some bounded absorbing t set B, as above, we have no guarantee that \t≥0φ (B) will be non-empty! Thus, in addition to proving that the PDE is well-posed, one typically needs to establish some smoothing properties to show that not only is there a set B that eventually \absorbs" all trajectories but also that this set is precompact in the function space on which we are working. Fortunately, this smoothing can often be proven for the sort of dissipative systems we discuss here and the systems described in the subsequent sections all have well defined attractors. The reduction of dimension of the problem which results from focussing one's attention on the restriction of the system to the attractor is an even more powerful 2 tool in the context of PDEs than ODEs. As we will see in subsequent sections, many physically interesting PDEs have attractors of finite (sometimes even small) dimension. Thus, the complex behavior of solutions of the PDE can be captured by the behavior of the solutions restricted to this finite dimensional set. In one sense this simply transfers the problem from the study of the infinite dimen- sional behavior of solutions of the PDE to computing the possibly very complicated structure of the attractor and the latter question remains an active and open area of research for many of even the most natural physical systems, but it is at least a finite dimensional question and one that is well suited to attack with the methods of dynamical systems theory. As mentioned above, in infinite dimensional systems, dissipativity requires more than simply the existence of a bounded absorbing set to insure that the system has an attractor. There are several additional hypotheses one can make to insure the necessary compactness - see [Hal88] or [Tem88], for example. However, we follow Robinson ([Rob01], p. 264) and define: Definition 1. A semigroup is dissipative if it possesses a compact absorbing set B. The drawback of this definition is that one must always check the compactness of the absorbing set. Given a dissipative semigroup φt on a Banach space X, the !-limit set of a set U ⊂ X is !(U) = fu 2 X j there exists u0 2 U; and ftng; tn ! 1; tn such that lim kφ (u0) − uk = 0 g : (5) n!1 A global attractor is then defined as ([Tem88], p. 21) Definition 2. A set A ⊂ X is a global attractor for the semiflow φt, if 1. A is invariant. 2. Every bounded set is uniformly attracted to A. That is to say, if U ⊂ X is bounded, then lim dist(φt(U); A) = 0 : (6) t!1 With these definitions it is relatively easy to show: Theorem 1. If the semiflow φt is dissipative it has a universal attractor. If B is a compact absorbing set for φt, then the attractor is !(B). t Proof. The proof is relatively straightforward. The sets A(t0) = [t≥t0 φ (B) are non- empty, compact and decreasing, and their intersection, which gives !(B) is clearly invariant under φt and one checks the attractivity property by contradiction. The details are in [Rob01]. 3 t t Remark 1. For a semiflow defined by a well-posed PDE such that φ (u0) = φ (v0) implies u0 = v0, one has the important corollary that when restricted to the attractor, t t φ actually defines a flow, not just a semi-flow. That is, if u0 2 A, then φ (u0) is defined for all t 2 R, not just for t ≥ 0. This is somewhat surprising since in general it is not possible to solve dissipative PDEs \backwards" in time. One may wonder how hard it is to establish that an infinite dimensional dynamical is dissipative in the sense of Definition1. Often the compactness needed in the definition follows in a relatively straightforward way from the same sorts of estimates that yield existence and uniqueness of solutions. Consider, for example, the family of non-linear heat-equations ut = uxx + g(u) ; 0 < x < L (7) with zero boundary conditions - i.e. u(0; t) = u(L; t) = 0. If one places mild growth conditions on the nonlinear term g this is known as the Chafee-Infante equation and was one of the first PDE to be systematically studied with the methods of dynamical 1 systems theory. If one assumes that the initial conditions u0 2 H0 (0;L), then the methods of semigroup theory readily show that the orbit u(·; t) with this initial condi- tion is relatively compact in this Hilbert space [Mik98], and one has an absorbing set. Hence the equation defines a dissipative dynamical system in the sense of the above definition. In this case the attractor is quite simple - for a special class of nonlinear terms, Henry showed that it consisted of the stationary solutions of the equation, and their unstable manifolds, [Hen81]. Of course, the converse is of this fact is also true - there are equations like the three-dimensional Navier-Stokes equation which are expected to be dissipative on physical grounds, but the fact that there is no proof that smooth solutions exist for all time for general initial data precludes proving that they are dissipative in the sense of the previous definition. The main example of an infinite dimensional dissipative system that we'll examine in the remainder of this review is the two-dimensional Navier-Stokes equation (2D NSE). These equations describe the evolution of the velocity of a two-dimensional fluid. While it may seem unrealistic to study two-dimensional fluid flows in a three- dimensional world there are a number of circumstances (e.g. the Earth's atmosphere) where this is a reasonable physical approximation. For further discussion of this point, see [Way11]. Physically, the NSE arise from an application of Newton's law for the fluid, namely d (momentum) = applied forces : (8) dt If u(x; t) is the fluid velocity, and if we assume that the fluid is incompressible so that we can take the density to be constant, the time rate of change of the momentum is given by the convective derivative d @u (momentum) = ρ + ρu · ru : (9) dt @t 4 The second term in this expression reflects the fact that the momentum of a small region of the fluid can change not only due to changes in its velocity, but also because it is being simultaneously swept along by the background flow.
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